International Bank for Reconstruction and Development
Development Research Center
Discussion Papers
No: 12
APPLICATIONS OF LOREN2 CURVES IN ECONOMIC ANALYSIS
N.C. Kakwani
August 1975
SOTE: Discussion Pnpcrs are prelin~inary materials circulated to stimulnte diecuosion and crirical -comment. References in publication to Discussion Papers 5ould be cleared with the author(s) to protect the tentative character of these
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The Lorenz c u r v e r e l a t e s t h e c u m u l a t i v e p r o p o r t i o n of i c c m e u n i t 6
t o t h e c u n u l a t i v e p r o p o r t i o n of income r e c e i v e d when u n i t s a r e a r r anged i n
a s c e n d i n g o r d e r of t h e i r income. I n t h e p a s t t h e c u r v e h a s been mainly u s e d
a s a c o n v e n i e n t g r a p h i c a l d e v i c e t o r e p r e s e n t t h e s i z e d i s t r i b u t i o n of
i n c o n e and w e a l t h .
The i n t e r e s t i n t h e Lorenz c u r v e t e c h n i q u e h a s been r e c e n t l y
r e v i v e d by Atlcinson [ 1 ] who p rov ided a theorem r e l a t i n g t h e s o c i r l
w e l f a r e f u n c t i o n and t h e Lorenz cu rve . He showed t h a t t h e r ank ing of
income d i s t r i b u t i o n s a c c o r d i n g t o t h a Lorenz c u r v e c r i t e r i o n is i d e n t i c a l
w i t h t h e r a n k i n g imp l i ed by a g g r e g a t e economic w e l f a r e r e g a r d l e s s of t h e
form o f t h e w e l f a r e f u n c t i o n of t h e i n d i v i d u a l s (except t h a t i t be
i n c r e a s i n g and concave) p rov ided t h e Lorenz c u r v e s do n o t i n t e r s e c t . Hou-
e v e r , i f t h e Lorenz c u r v e do i n t e r s e c t , o n e can a lways f i n d two f u n c t i o n s
t h a t w i l l r ank them d i f f e r e n t l y . Das Gupta , Sen and S t a r r e t t [ 2 ] have
shown t h a t t h i s r e s u l t is i n f a c t more g e n e r a l and does n o t depend on t h e
a s sumpt ion t h a t t h e w e l f a r e f u n c t i o n s s h o u l d n e c e s s a r i l y b e a d d i t i i e .
I n t h e ' p r e s e n t pape r t h e Lorenz Curve t echn ique is used a s a
t s o l t o i n t r o d u c e d i s t r i b u t i o n a l c o n s i d e r a t i o n s i n economic a n a l y s i s . T h e
G n c e p t o f Lorenz cu rve h a s been ex tended and g e n e r a l i z e d t o s tudy t h e '9 L
r_e l a t i onsh ips among t h e d i s t r i b u t i o n s of d i f f e r e n t economic v a r i a b l e s . D.e *
I! S n e r a l i z e d L l ~ r e n z c u r v e s a r e c a l l e d c o n c e n t r a t i o n c u r v e s and t h e Lorenz .
c u r v e i s o n l y n s p e c i a l c a s e o f such c u r v e s , v i z , , t h e c o n c e n t r a t i o n c u r v e
1 / f o r income.-
I / P r o f e s s o r Mahalonobis [ 6 ] used c o n c e n t r a t i a r i c u r v e s t o d e s c r i b e t h e - consumption p a t t e r n f o r d i f f e r e n t commodities based on t h e Na t iona l sample Survey Data . See a l s o Roy, Chakravnr ty and Laha [ 7 1
Sect ion 2 gives t h e de r iva t i on of the Lorenz curve. Some theorezs
r e l a t i v e t he concent ra t ion curve of a func t ion and i ts e l a s t i c i t y a r e
provided i n Sec t ion 3. These theorems provide t h e b a s i s t o s tudy r e l a t i on-
sh ips amon; t h e d i e t r i b u t i o n e of d i f f e r e n t economic va r i ab l e s . Appl icat ions
of the t heo rem a r e discussed i n Sect ion 4.
2 . THE WRENZ CURVE
Suppose t h a t income X of a family is a raildam v a r i a b l e with
p robab i l i t y dens i t y func t i on f ( X ) . Then t h e d i s t r i b u t i o n func t ion F(x)
is defined a s :
and t h i s f unc t i on can be i n t e rp re t ed a e t h e proport ion of fami l ies having
income l e s a than or equal t o x.
I f i t is us& that t h e mean E(X) - N of t h e d i e t r i b u t i o n
e x i s t s and X > 0 , then t h e f i r s t moment d i s t r i b u t i o n func t ion of X is - dc f incd as: ;
The Lormz curve i a t he r e l a t i o n e h i p b e t v e e n P(x) and F l ( x ) . The - . * grap%gf t he curve i e repreeented i n a u n i t square. The equatioxr of t h e -.
l i n e P1 F l a c a l l e d t h e e g a l i t a r i a n l i n e and i f the Lorenz curie c o i n c i d e s
with thim l i n e i t impl ies that each family receive8 the same income.
Tlie most widely used measure of i n e q u a l i t y i s G i n i ' s Index which
is e q u a l t o twice t h e a r e a between t h e Lorenz curve and the e g a l i t a r i a n
l i n e . I t can be w r i t t e n a s : -
a0
and i t v a r i e s from zero t o one.
3 . THE CGNCENTRATIGN CURVES
Let g(X) b e a cont inuous f u n c t i o n of X such t h a t i ts f i r s t
d e r i v a t i v e e x i s t 8 and g(X) 2 0 f o r X > 0 . I f E [g(X)1 e x i s t s , then - one can d e f i n e :
so t h a t F [ g ( x ) ] i s monotonic i n c r e a s i n g and F1 [g (o ) 1 0 and PI [g(m) 1 11. 1
The re l . a t ioneh ip between F1 [ g ( x ) ] and F(x) w i l l be c a l l e d t h e concenkra-
. ? t f o n curve of t h ~ e f u n c t i o n g ( x ) . & - - - . I t can be se )n t h a t t h e Lorenz curve of income x i s a s p e c i a l *
I - case of the c o n c e n t r a t i o n curve f o r t h e f u n c t i o n g ( x ) when g(x) - x.
The above g e n e r a l i z a t i o n of t h e Lorenz curve was suggested by
Profesrsor P.C. 4lnhalanobis t o d e s c r i b e t h e consuroer behav iou r p a t t e r n v i t h
r e s p e c t t o d i f f e r e n t commodities.
The r e l a t i o n s h i p between 5 [ g ( x ) ] and P (x) w i l l be c a l l e d the 1
r e l a t i v e c o n c e n t r a t i o n c u r v e of g(x) wi th r e s p e c t t o x. S i m i l a r l y . l e t
* g (x) be ano the r cont inuous f u n c t i o n of x , then t h e graph of F1 [ g ( x ) ]
* F1 [ g ( x ) ] will1 be c a l l e d t h e r e l a t i v e c o n c e n t r a t i o n curve oL g(x) v i t h
* r e s p e c t t o g (x) . Let 0 (x) be t h e e l a s t i c i t y of g (x ) v i t h r e s p e c t co
8
x , then:
where g l ( x ) ,is t h e f i r s t d e r i v a t i v e of g ( x ) . *
S i m i l a r l y denote (x) a s t h e e l a s t i c i t y of g (x) with r e s p e c t 8*
We can now s t a t e t h e fo l lowing theorem:
Tli60REV I : The c o n c e n t r a t i o n curve f o r t h e f u n c t i o n g (x ) w i l l l . ie above - *
(below) t h e c o n c e n t r a t i o n curve f o r t h e f u n c t i o n g (x) i f
0 ( x ) i s l e e s ( g r e a t 5 r ) than 0 (x) f o r a l l x > 0 : B 8* -
Proof of the Tkeorw 1
Using t h e equa t ion (3.1) we o b t a i n :
- 5 -
vhich g i v e t h e s l o p e of t h e r e l a t i v e c o n c e n t r a t i o n curxVe o f g ( x ) w i t h
* r e s p e c t LO g ( x ) a s :
The equa t ion (3.6) i m p l i e s t h a t t h e r e l a t i v e c o n c e n t r a t i o n cu rve is monotonic
i n c r e a s i n g . S i n c e t h e c u r v e must p a s s through ( 0 , O ) and ( 1 , l ) i t f o l l w s
t h a t a s u f f i c i e n t c o n d i t i o n f o r El [ g ( x ) ] t o b e g r e a t e r ( l e s s ) than
* F1 [ g ( x ) ] is t h a t t h e c u r v e be convex (concave) from above. To e s t a b l i s h
c u r v a t u r e we o b t a i n t h e second d e r i v a t i v e of Fl [g (x ) ] w i th r e s p e c t t o
t h e s i g n of t h e second d e r i v a t i v e is g iven by t h e s i g n o f n ( x ) - n (x) . g g*
Thus t h e second d e r i v a t i v e is p o s i t i v e ( n e g a t i v e ) i f n i s g r e a t e r ( l e s s ) R
then f o r a l l x . Hence t h e c o n c e n t r a t i o n cu rve f o r g ( x ) is above & * *
(below) t h e c o n c e n t r a t i o n c u r v e f o r g (x) i f rl (x) i s l e s s ( g r e a t e r ) g . P
than (x) f o r a l l x 7 0 . i3
' S * Let g (x ) = c o n s t a n t f o r a l l x > 0, then t h e e l a s t i c i t y n (x)=O -
L * git - an* F1[g ( x ) = F(x) which is t h e equa t ion of t h e e g n l i t a r i n n l i n e . Thus*
m . we have t h e fo l lowing c o r o l l a r j .
.';.~G~L.t.?? I : z h e c o n c c n t r a t i a n c u r v e i c r :hc ~ J ~ I C L : ~ ? g ( x ) v i l l 'l:e z b c . ~ ~ .
(below) t h e e g a l i t a r i a n l i n e if (x) is l e s s ( g r e a t e r ) thz.-. g
zero . --
-. The proof of C o r o l l a r y 1 !s a l s o d i v e 0 by ?.;.;, , r l zXravar t i a c d
* i a h a [ 7 1. Next we assume t h a t g (x) = x s o ti-.at - x ) = 1 and t h e
,4 * * c o n c e n t r a t i o n c u r v e f o r g (x ) i s ~.c-+~ the Lzrens Tcr t h e d l s t r i b u t i o ~
x . I c f o l l o w s f rom t h e C o r o l l a r y i t!:clt tk.- L c r e n z c a r v e f o r x l ies belcr.
t h e e g a l i i a r i a n l i n e and t h e r e f o r e t h z cT:~;c I s ccncnve fro2 abme. F u r t h k r ,
from Theorem 1 we have t h e f o l l o w i n g C o r o l l a r j .
CO?CLLARY 2: The c o n c e n t r a t i o n c u r v e f o r t h e f ~ : i c " -7 - . g ( x ) l i e s above -----
(below) t h e L o r e n ~ c u r v e f o r t h e d i s t r i b u t i o n s f x i f -
I ~ ~ ( x ) i s l e s s ( p r e a t e r ; thnn t in icy fo r a= x > - 0 .
I f t h e f u n c t i o n g(x) h a s t h e u n i t e l a s t i c i t y f o r a l l x - > 0 , t':e
second d e r i v a t i v e f o r t h e r e l a t i o n c o n c e n t r a t i c n o f g(x) w i t h r e s p e c t t o
x w i l l be z e r o which i m p l i e s t h a t s l o p e of t h e r c l a t i . : ~ concent ra t l ion curve
w i l l h e c o n s t a n t For a l l v a l u e s of x . S i n c e t h e cu rve s : ~ s t pns s through
(0 ,Oj and ( 1 , l ) i t means t h a t t h e r e l n t i v e z o n c e n t r i ? : ! ~ ~ cl t g(x) wi th
r e e p e c t t o x , c o i n c i d e s w i t h t h e l i n e ( 2 , G ) and (I,! i . l i ~ n c e
F1 [g(x)] - F ( x ) f o r a l l x ; k-hjch pr- jvco ti?c f o l l o b - f n g : 1
.a CCROLLARY 3: The c o n c e n t r a t i o n c u r v c f o r g ( x ) cci r .c idcs w i t h t h e Lore- . -- c u r v e f o r i f r? ( x ) - i o r A ; I ;. ; of X .
g ---- e- - I t shou ld be p o i n t e d o u t t h a t i!le ~ c n r r n i r a t ? r i . For g ( x ) ? s :;.:
1 7 the same t h i n g a s t h c Lorenz cur-Jc i i t r 7 ':,, . v ( -. : i.c.- l i o , ; t ~ :kc
c o n d i t i o n under which bo th a r e i d c n : i c ~ ~ .
let y =: g(x) be a random variable with prokability density
* * f~ r . c : i on f ( y ) and the distribution function F ( y ) , and if rean of
v exists, the first moment distribution funstior. cf y I s g iven by :
-
0 * * then [ F (y) , F (y ) ] is a point on the Lcrenz curve f c r g(x ) . The
1
following theorem gives the conditions uader which:
* * F (y) = F(x) and F1(y) a Fl [ 6 ( x ) 1 (3 .9 )
fcr all values of x.
TI!E(?PE?':' 2: - If g(x) &strictly ~onotonic and has a continuous derivative
g' 0:) > 0 for all x, - then the concentration curve for g ( x )
coincides with the Lorenz curve for the distribution of g ( x ) . -
t>oo f of the Theorem 2
Under the oeeumption that g ( x ) is strictly conotonic and has a
cor,tinuoue non-vanishing derivative in t1:c region : , :he probo~bility
density function of y is given by II
* f (Y) E: f [ h ( y ) j 1 h f ( y ) ( (3 . ; 3 )
- - where x - h ( y ) is the solution of y - g ( x ) . ' J
L - * , 8 . Let US now consider the gl-apt1 of F (x) -JC F f g ( > r j ] which h a s ti:< C i -
s l o p e
v h i c h ion u s i n g (3 .10) becomes one if h l ( y j > 0 . h ' ( y j is obv ious ly
g r e a t e r t han z e r o f o r a l l y . F u r t h e r s i n c e g l ( x j > 0 and t h e cu rve m u s t
* pass through (0.d) and ( 1 , l ) i t i m p l i e s t h a t t h e c u r v e F [ g(x) ] .vs
F(x) v h i c h h a s c c n s t a n t s l o p e one must c o i n c i d e wich t h e l i n e p a s s i n g -
* through (0,O) and ( 1 , l ) . Hence F [ g(x ) ] = F ( x ) .
* S i m i l a r l y i t can b e proved t h a t t h e g r aph of F, [ g ( x ) ] v s F1 [ g ( x ) ]
1
has s l o p e one i f h t ( y ) > 0 . S i n c e t h e c u r v e p a s a e s t h r o u g h (0.0) and
( i , i ) , i t must c o i n c i d e w i t h t h e s t r a i g h t l i n e j o i n i n g (C,O) and ( 1 , l )
* which i m p l i e s F1 [g (x ) 1 = F1 [ g ( x ) ] . T h i s p r o v e s t h e t h e o r e a .
I I T 1: The f u n c t i o n g(x ) i s s a i d t o be Lorens s u p e r i o r ( i n f e r i o r )
* t o a n o t h e r f u n c t i o n g (x) i f t h e Lorenz c u r v e f o r g(x )
* l i e s above (below) t h e Lorenz c u r v e f o r g (x) f o r a l l
I t f o l l o w s from t h e d e f i n i t i o n of Gin i- Index t h a t t h e d i s t r i b u t i o n
gene ra t ed from f u n c t i o n g ( x ) w i l l have lower ( h i g h e r ) v a l u e of Gini- Index
* thnn t h e d i s t r i b u t i o n g e n e r a t e d from g (x ) i f g(x) l a Lorenz e u p e r i o r
* ( i n f e r i o r ) t o g (x ) .
* C I f t h e f u n c t i o n s g ( x ) nnd g ( x ) a r e s t r i c t l y monotcnic and
have c o n t i n u o u s d e r i v a t i v e s s t r i c t l y g r e a t e r thnn z ~ r o , t hen from Theorem - - 2 i t f o l l owe t h a t t h e i r c o n c e n t r a t i o n c u r v e s co . inc ide w i t h t h e i r r e s p e c t i v e J
Q
Lorenz c u r v e s . Then u s i n g Thebrem 1 we o b t a i n t h e fol!\wing C o r o l l a r y . - - E I * Js
1
fcFc.-!*lf?Y 5: -- I f t h e f u n c t i o n s g(x) a$ g (x) ore e t r i r t l y ~ o n o t o n i c -- and hnve c o n t i n u o u s d e r i ~ n t i . ~ , e s ~ t r i c t l v g r e a t e r t h a n z e r o ,
R
it re^^ go : ) is r i r ( i f . ' 2 (x,)
i f q (x) is l e s s ( g r c n t e r ) t han 9 ( X I f c r a?: - S ----- *
* ; \ ~ a i : ~ if we p u t g ( x ) = x S O t h a t q X ) 1 r k e 1 Ccrcll.:r:;
g * -
5 l e a d s t o t h e i -o l lowlng C o r o l l a r y .
:;,L.,;LL/:?)' 6: I f g ( x ) i s s t r i c t l y n o s o t o n i c and i:zs a c 3 n t ! ~ u o u s c e r i v a - - - -
t i v e g ' ( x ) > 0 f o r a l l :i: t h e n g(x) Is ;,c:enz s u s e r l u r
( i n f e r i o r ) t o x - i f i ip(x) i s l e s s (2ri-;iir t:ioo r, ( x ) g
f o r a l l x > O . -- -
C 2: The c o n c e n t r a t i o n i n d e x f o r gix) l e f l n e d a3 9 n e z i n 3 . i ~
t w i c e t h e a r e a u n d e r t h e c o n c e n t r a t i o n c E r v e f c r_ g ( x ) .
I n ollr n o t a t i o n , t h e c o n c e n t r a t i o n i n d e x f o r g(x) i s g i v e n b y :
1)
C g = * - 2 ' F1 [ g ( r ) ] f ( x ) dn. I
1 t 1!1 co h c not.ed t11nt j f g ( x ) = corlr;t.nnt., t h e c o ~ ~ r ~ ~ n : r , ~ : i c ~ n c r i r v D coi-.c5,.t2.
w i t h t h e c p , a l t l : a r l a n l i n e s o t h a t C - 0. I f ( x ) : -I i s d n y
c o n s t a n t , t h e n t h e c o n c e p t r a t i o n i s e q u a l t o t h e Gini-Inr!t .x of x . F\:r '&her,
i f g ( x ) 0 f o r a l l x , t h e n C is n l v z y s p o s i t l : . ~ ' 8 i n d v l l l bc: c q : i n l r; R
t h c GlnL- Index o f t h e E u n c t l o r ~ g ( x ) . F i n a l l y i f b ( x ) C) fo r a l l X ,
t h ~ n t h c c - o n c e r r t r a t i o n c u r v e f o r g ( x ) i s a b o v e t h e er,,+l! t ~ r ; a n i:r.c <=r.d - C wlll Se e q u a l t o m i n u s t i m e s F
k :c -,.--..-, -.zL:L.-L,4 3: ~f g ( x ) = 1 g i ( x ) s o t h a t E !a(:.:)] - t 5 ( ' X I ]
1: 1 i i= 1
where E i s t h e expec t ed v a l u e ope=r, : t en :
I Y C ~ f of the Tnecrem 3
k 3 1 S u b s t i t u t i n g g ( x ) = gi(x) ia (3.1) g'.vtJ:--
%=I
N o w Pi [ g i ( x ) ] i s g iven by:
t h i c h on s u b e t i t u ! t i n g i n (2.13) g i v e s t h e r e s u l t s t a t e d i n T h e o r a 3.
. E x ; the;. Le t g(x) = n+bx E O t h a t E [ g ( x ) ] = a+bu , ~ i . - - - -
g(x) cnn b e t r e a t e d R B t h e Bum of two f u n c t i o n s , v i z , a and b x . Hence f r o 2
TIleorem 3 ve o b t a i n :
Because ttt! c o n c c n t r n t i o n c u r v e f o r a c o n s t a n t funce on c o i n c t d e s w i th t h e - I egalitarl& l i n e . The e q ~ a t i o n (3 .16) con a l s o be writttn ;ri:
31 The i n t e r c h a n g e of eumriatiorr s i g n 2nd -- k is f i n i t e .
S i n c e F ( x ) .- > F1(x) f o r all . x i t i ~ ~ p l i e s t h a t t h e c o l : c e r . t r c ~ i . n cur-;? i c r
R 1 i n c a r f u n c t i o n (a i- bx) l i e s above ( ' te low) t h e Lorc:;z cu rve f o r u i f
n is g r e a t e r ( l e s s ) t h a n z e r n . F x r t h e r i f b>O , t h e function g < x ) = a -
bx is a monotonic i n c r e a s i n g f u n c t i o n of x , frox Theor& 2 5~ f u l l - ~ s thz:
t h e ~ o n c e n t r ~ a t i o n c u r v e f o r (a + bx) c o i r c i d e s w i t h the Sorenz CLL:-ve of
f u n c t i o n (a + b x ) . Thus w e have t h e f o l l o c i n g c o r o l l a r ; J .
, - TChOI.Ur7Y 7: Tf b > 0, t h e n t h e l i n e a r f u n c t i o n ( a 4 b u j -15 Lorer.z --
s u p e r i o r ( i n f e r i o r ) t o x if a is g r e a t e r ( l e s s j t h a ~ .
ze ro .
k k 1 - . 3 1 .
I titi,,<Cd 4 : -- - I f g ( x ) - 1 gi(x) SO that.+ E[g,x)] = 1 K f s . (x)] , i=l 1 in1
t h e n : - k
E t a ( x ) l cg - 1 ~ [ g ~ ( x ) l c g i ( 3 . 2 1 ; in1
i h e r e Cg and Cgi a r e c o n c e n t r a t i o n i n d i c e s f a r g ( r ) g i ( x ) ,
r e s p e c t i v e l y .
1'1-oc~ f of th Theorem 4 : --
S u b s t i t u t i n g (3.13) i n (3.12) g i v e s : . r
m k
, , i ( x F i g i j f X x ( - . - 2 ,
1 - 0 '2 L
w!>ich on i n t e r c h a n g i n g t h e s-tion and i n t e g r a l ~ i g ; : f i 5rrc L z ~ 9 :
a w L U
k 2 .. I
3 . - - 5 " 1 - - 1 E[p.i(~p:) j 1 - i f (u):;:< E I g ( x ) ! i
- . L
>
f iow C 8 1 is d e f i n e d o f i :
-12-
L u
F [g , (x)] f (x ) c ix C g l - l - 2 \ 1 ( 3 . 2 ~ ; 0
- k S t t S s t i t u t i n g ( 3 . 2 4 ) i n (3.23) and u s i n g t h e f a c t that E[g(xjj = 1 E [ z (x,:
i=1 i
g i v e s t h e r e s u l t (2 .3Gj . T h i s proves t he theorex .
L e t u s a g a i n assur;le t h a t g(x) = a+bx so that E[g(x ) ] - a+Ep .
I f b > 0 , g ( x ) 13 a monotonic i c c r e ~ s i c g fcnct:cn, t h e r e f c r e the c o n c a -
t t a t i o n index f o r g ( x ) w i l l be saze as t h e G in i- index of the f lalnct im.
N w us ing t h e f a c t t h e Gini- Index of a constant is z c - o , and t h e
Gini- Index of bx is same a s t h e Gici-index of x , i t foilcws from T h e o r c
* where G i s t h e Gi r i - Index of x arld G i s t h e Gin i- Index of t h e l i n e a r
f u n c t i o n x ( a + bx) . We have the f o l l c d i n g c o r o l l a r y .
L'CR(?LIARY 8: I f G i s t h e Gini- Index of n random v a r i a b l e x, then ihe ---- - *
Gini- Index G of a l i n e a r f u n c t i c n ( a 5x) f o r b > 0 -- -- . c
is g iven by:
- where E ( x ) - P , * s
* * 'e In t h e above Corollary i f a - 0 , G = G w h f c h !.npifei? r:-,st i f all incc-,__.
a r r n r l l t i p l i r d by a anme c o n s t a n t , then t h e
* Further, G is iess ( g r e a t e r ) t h a n G I f
I n t h i s s e c t i o n w e s h a l l c o
n
s i d e r s o ~ ? :lf i h e a::;lic,;tic!zs of :he
4 / t ? i ~ c r c z q g i v e n i n t h e i n s t s e c t ion .-
- . . . I f g(x) i s t h e e q u a t i o n of Er;gal Ci;rvt'. 91 z .-r~..;nc; t-'. -, , then < r
fo!lor;s f r o n C o r o l l a r y 1 a n d 2 thz: i f i t s concen t r s r i r in zclrve l i es a b o v e
t h e e g a l i t a r i a n l i n e , i t i s a n i n f e r i o r co i - l -okJ l t> , iL : o c r t n t r a t i u n
c u r v e l i e s be tween t h e t o r e n z c u r v e of x an3 r;!c k s n l 3 t i i r : n n i i n e , 1: i s
a n e c e s s a r y commodity a n d i f t h e c o n c - . n t r a t i o n curlre lit:; r e i o 1 t h e L o r e n z
c u r v e , t h e commodity i s l u x u r y .
4 . 2 Ccnsmption and .Saving Filnctio~ro
I n t h e K e y n e s i a n c a s e t h e c o n s u m p t i o n is r e l z c e d t o i n c o x e e i t h e r
l i n e a r l y o r c u r v i l i n c ~ r l y . L e t u s f i r s t a s s n n e t h a t t h e r t l s : ! o n be i i n e a ; :
* .
v h e r e 3 i s t h e r l a rg 'na l p r o p e n s l i y t o cansuz:* f:::ci - i ;- :!-:c. : !spor. -~F. le
jnconr a n d c is the conGumption e x p e n d i t u r e of a n inr'.i:.i:;;:n-. S i n c e n
and r a r e : r e n t e r t h a n z e r o , i t f o l l o u s f r o n C o r o l 2 . z r - y 7 ~!i:>t t h e ; E T : - c : . ~ - - - 'i L
c c r i s u r ~ p t i o n e y e n d i t u r e 1s moLe e q u a l l y d i s t r ! ' 8.:t.i : t -.- " . r c - r t o r , ~ ? - d i s p o s a b l e income.
-
!+/ .'.!any n o r e a p p l i c a t i o n s o f t h o ti;carLl:a:; w!; l i-.;: .-:li ;-;:,-: . I ' ' - ,. . , 3 lc~::::-;~:<:L.:.. :.onoqraph w h i c h i s ;:nc!e:- p rep~r ; \ : t : ; i ? .
~+..f..Lch a g a i n f r o n C o r o l l a r y 7 i m p l i e s t h ? t t h e persGr.e7 sr ;vir ,zs -d i l l b e c o r e
uEequal ly d i s t r i b u t e d t h a n t h e p e r s o n a l d i s p o s a b l e income provzded t h e
a a r g i n a l p r o p e n s i t y is l e s s t h a n oxe.
Let U E now i n t r o d u c e t h e r a t e of i n t e r e s t a s a n ac!di:ional v a r i a b i ~
i n t h e savings f u n c t i o n ( 4 . 2 . 2 ) :
& e r e r i s t h e r a t e of i n t e r e s t . I f 8 < 1, t h e n f ron C c r o l l a r y 8 w e
o b t a i n :
v h e r e G and G a re G i n i - I n d i c e s of d i s p o e a b l e income and r,avings, r e s p e c - 8
t i v e l y . i n t h e mean d i s p o s a b l e incoine and us i s t h e cean s a v i n g x.iiick.
? a g iven by:
. i
h
Dl f r e t e n t i o t i n g ( 4 . 2 . 4 ) w i t h r e s p e c t t o r g i v e s :
C
vh c h l e o d r t o t h e c o n c l c s i o n t h a t h i g h e r t h c i n t c : . - i : , ? T c , xsre cr, ';ai r > -
5 e t h e d i s t r i b u t i o n of s a v i n g s . This concltlo!on i:. of cp .a r r r . 5:iseZ CI; :':,,
~ s s : m p t i o n t h a t th2 i n c r e a s e i n t h e l n t c r c , s t ro:c ::L,c. - - ' .4 L I ter "-- C r e
d i a t r i b u t i o n o f t h e d i s p o s a b l e i ncane .
I i I
-.16-
1 i I . .
? . I J-ZZXIILZ i?! un T n i I . ~ t i m " a r y i ' c s z ~ , ~ ~ I
1 Consider a n econcny i n which p r i c e s and prodcc~i- : i :y a r 2 r i s i n g 2:
dnnclal r a t e of 100 p and 100 s p e r c e n t . Sus-,ose t h e ~ ~ I Z P T C S of a l l i n c c a e
, . :nits a r e i ~ c r e a s i n g i n t h e same p r o p o r t i o n . Then ir.coc.c i?f a u n i t a f t e r t
where x is t h e i n i t i a l income. Le t t h e t a x f u n c t i o n 1 ;s :
t hen t h e t a x c o l l e c t e d a t t ime t from a n incorre u n i t ult!: i n i t i a i income ;;
w i l l b e :
l a t h c mean t n x pa id at t ime z e r o , then t h e ccnn t a x p o i 2 n : r:me t v i i l be -
b h i c h g i v e s t h e ave rage tax r a t e z t t i c . e
L;,CX c ;:(Y) = ,. y - [ r . ( t ) ] = , ~ ( t ) . T'llus, i f t i le t a x e s tire ; r o g r c s s i v e
: > 1 , t!le s -~ : ragc t a x r a t e w t l l ir!crcl:!se (c iecret ise) c7,.cr ti.-* i f ? , s a r p
g r e a t e r ( l e s s ) t h a n z e r o b u t l e s s t h a n one i n a b s o l u t e v a i u e .
The d i s p o s a b l e income a t t i n e t of a u n i t havir2g i a i t l z i . i a c o ~ i z
x is x ( t ) - T { x ( t ) ] a n d , t h e r e f o r e , a p p l y i n g Theoi-er 3 xc. o:, tain:
where
q ( x ) = $ 1 a x6 f (x) d x
i s t h e p r o p o r t i o n o f t a x p a i d by i n c o n e u n i t s h a v i n g i n c c z z l e s s t h a n o r e q u a i
* L C J x a t t i m e z e r o and Ft (x ) is t h e p r o p o r t i o n o f t h e
* t h e d i s p o s ; l b l c income o f t h e same i n c o n e u n i t s a t Lime e . , ( r ) i s t l ~ e
ne3n d i s p o s a b l e income a t t i m e t :
* t , ? t p ( t ) = { ( l f p ) ( l + s j } p - {( l . l l1 ) ( l L : j j (: (4 .L.G) - -
Tl:e ecluat i n n ( 4 . 4 . 7 ) simplifies t o :
I f t h e t a x f u n c t i o n i s p r o g r e s s i v e , i . e . 5 > L, t h e n f r o n
C o r o l l a r y 2 , F, (x ) > q (x) f o r a l l x u h i c h f rom (1.6.10) i z p l i s s that .. the c o n c e n t r a t i o n c u r v e f o r t h e d i s p o s a b l e incor-e ar: ti-ze : I s h i g h e r
-
t h a n ? h e L c r e n z c u r v e f o r i n c o ~ e . F u r t h e r , i f ::?e zarg::,.?l t c x r a t e :Ls
l e s s t h a n o n e , t h e d i s p o e a b l e i n c o a e i s a n o n o t o n i c i n c r e s s i c g f u n c t i o n
o f x v h i c h f r o n T h e o r e n 2 i n p l i e s tha t t h e 8ccr.cectrat?o?. C ' I N ? f o r :he
d i s p o s a b l e income a t t i m e t c o i n c i d e s w i t h i ts L o r e n z z : r v e . T h u s f c r
n p r o g r e s s i v e t a x s y s t e m t h e a f t e r t a x i n c o ~ e at t i r e r ? s m r c e q u a l l y
d i s t r i b u t e d t h a n t h e b e f o r e t a x i n c o r e .
Di f f e r e ~ n t i a t i n g ( 4 . 4 . 1 0 ) w i t h r e s p e c t t o p gi3;es :
A g a i n , i f t h e t a x s y s t e m i s p r o g r e s s i v e 6 > 1 and i (x) > q ( x ) 1
which i m p l i e s t h e r i g h t - h a n d s i d e o f ( 4 . 4 . 1 1 ) i s p ~ s i t ! - . ~ e ~ n c i , t h e r e f o r e , ,is
p i n c r e a s e s t h e L o r e n z c u r v e f o r a f t e r - t a x income d i s t r i l u ~ i o n v i l i s h i f t
c p v a r d . S i m i l a r l y , i f t h e t a x s y s t e m i n r e g r e s s i v e , 6 1 f i n ] q k x )
t h e r i g h t- h a n d s i d e o f ( 4 . 4 . 1 1 ) is a g a i n p o u i t i v e . The !.%crr,i?z c u r v e ~ i h i f t s L
upvnrJ no p i n c r e a s e s . Thus we c a n c o n c l u d e t h a t t h e i n f l n e l o n decrease^
t h e a f t e r t a x i n c o m e- i n e q u a l i t y f o r b o t h p r o g r e s s i t c and :r;;r:.r:3ivc t a x
systerno p r o v i d e d t h e b e f o r e t a x d i s t r i b u t i o n i s n o t affected by i n f i a r i o n . - The a b o v e c o n c l u s i o n i o v a l i d ably i f t h e ' i sx ; .~ a c e c a t 3 d j u s ; e a
t o i n f l a t i o n . Su,ppose w e c h a n g e t h e t a x r a t e s every yeAr kj- b c s p i a g t
".- c c n e t n n t b u t c h a n g e t h e p a r a m e t e r t . i n e t a x f ~ m c ~ ' l a - . t i t -2r-c i <:an
t h e n b e * n i t t e n ae:
v h e r e a - 3 a t t = 0 . Then t h e mean t a x a t t i n e t will be: t
a t
~ ( t ) = [ (1 + p ) (1 + 9) j 6 t Q ( 4 . h . l j )
and , t h e r e f o r e , t h e a v e r a g e t a x r a t e b e c m e s :
Suppose we a d j u s t a eve ry y e a r s u c h a way t h a t t!.e r a t i o of t
t o x t o income rer0,aine c o n s t a n t . Then from (4 .4 .14) i t can be seen t h a t
v h i c h means a is t o be r educed eve ry y e a r i f t h e tax f ~ x . c t i o n i s t
p r o g r e e s i v e and f o r a r e g r e a b i v e t a x f u n c t i o n a s h o u l d he increosec i . t
Nuw ue ing (4.4.15) I n (4.4.10) g i v e s :
- - * which implieu t h a t d P t ( x ) / dx I 0 . Thus we conclude: thn t t f t h e : n x
3 - f u n c t i o n i s a d j u ~ t e d eve ry y e a r such n \ jay t h a t t h e tax-lncrtr,.e r a t 2 0 i s *
* I concirsnt rvery y e a r , t h e n t h e i n f l a t i o n w i l l n o t ctiiinge t k r nf:er ?ax incs:,
d i ! l t r i b .d t i on f o r any tax sys t em p r o g r e n s i v e o r rcgressivc.
L e t :
* Ct
a Cin i- index of t h e a f t e r - t a x d i s t r i b u t i c n 3t
t i n e t .
C = Gini - index of b e f o r e- t a x incone a t t=0 .
- C o n c e n t r a t i o n index of t a x e s p a i d a t t i n e
ze ro .
f rom Theorem 4 we o b t a i n :
vh i ch g i v e s t h e e l a s t i c i t y of t h e Gin i - Index of t h e a f t e r t a x d i s t r i b u t i o n
wi th r e s p e c t t o i n f l a t i o n r a t e a s :
We can now compute t h e Gin i- index and t h e e l o c t i c i t y of t h e Gin i -
index v i t h r e s p e c t t o i n f l a t i o n r a t e . The s o u r c e of d a t a use6 f o r t h i s
purpose i s t h e A u s t r a l i a n Taxat lolf S t a t i e t i c s f o r t h e a s sc s s r r sn t yea r 1971-;2
(Income t n x y e n r 1970-71). lT.e d a t a n r e s v o i l n b l e i n gronped form. The
incoae c o n e i d e r e 2 i a t h e a c t u a l income f o r i n d i v i d u a l t n x pnyc r s less t h e ' ? -
e x p e n d i t u r e i n c u r r z d i n g a i n i n g t h a t income. - ~ i n e - - i n & of b e f o r e t a x i n c ~ m e vat3 conprlteC t o b e . 3 4 5 6 and f o r
t h e t a x p a i d t h e c o n c e n t r a t i o n index w a s . 5 4 1 9 . The t a x funcl . ion was
5 / e a t i ~ m t e d t o be : -
5 / The weigh ted r e g r e s e i o n method was uaed t o e s t l r i t c t h e ta:: f u n c t i o n . -
l o g T = -6.2064 + 1.583 l o g x ( L . 4 . i > ;
-.h,!re .,: rt,srcsc-:~ts i n c o ~ r a n 2 T t~3:ics. 'The squsre.? c u r r e l n t i o r , bc.:.;c_en
~ ~ t : ; r . j t ~ d a n d a c t u a l v a l u e s of T ~ 3 s conputed t o be .99.
~ t b l c I j r e s e n t s t h e C i n i - i n d e x of t h e a f t e r - t a x incose a n d i ~ s
e l a s t i c i t y v i t h r e s p e c t t o t h e raLe cE i n f l a t i o n . I t i s t o 5 2 noted t h e
Gin[- index is q u l t e s e n s i t i v e t o t h e i n f l a t i o n a sd t h e s e n s i r i v i t y i n c r e a s s s
wi th t h e r a t e of i n f l a t i o n an2 a l e o over t i z e .
Table 1: GIXI-LWEX OF THE kT@i TAX LNCOME AND ITS ELASTICITY WITH RESPECT TO IWPLirTiOH RATE
* u 9 1973 - 1974
Glni- Index l l a s t i c i t y
.0201
.3138 .0109
1972 - 1973 Gini- Index E l a s t i c i ~ y
Rate a t I n f l a t i o n
.31?5
.31G3
I
.0067
0.0000 1
. , I24 - .0071
- -10
1970 - 1971 1971 - 1972
0 . 0 0 1 .3110 -, 0021 .3096 1 -. 0047 .JOB1
C i n i - I n d u E l a s t i c i t y Gini- lnden E l a s t i c i t y
I .3124 0 .00
.3124 1 0 .00
I I 1 ,3121; .3105 - .3080 . 'Q65 -. 01 2 3
I 0 .3124 O.oO j . ) I17
I
I I
I
.0143
.0074
.3141
. 3 i 2 9
I
.3086 I
r a
. 3 1 2 4 1 0 .00 .309k -. 0074 . 3 0 h l -.0165 .7021+ -.0:<7Y r d 1 10 1
! I !
i .3124 1 0.00 .3083 -. 0110 . 3075 -. 0 2 1 , . 7Qi i l I -.01+55, [ i 5 i I 1 I 1
_ -- . -. -i I----- --- ---a -- i
.0045
0.0000
.1124 -312G i
0.0000 .31iO
I .0076
.0038
.0022
. ? I 57
.3134
.3124
S u ~ p o s e t I ~ e t o t a l f ami ly i nzone x i s i i r i ~ c e n a s :he sdn, of n f z c t r s
i ~ l c o r n ~ s x , , Y.~, . . . . x t h e n from Thecrem 4 , ~ - 2 o b t a i n A n '
C i is t h e c o n c e n t r a t i o n i n d e x of t h e i - t h f & z f o r i n c c z : ~ ceni3onent b-hich b.2;
:can income . 111s e q u a t i o n e x p r e s s e s t h e Gin i- index of t i e t o t a l f ami ly
Income aa t h e weighted a v e r a g e of tlrc c o n c c l l t r a t i o n i n d i c e s of each, f a c t o r
income component, t h e we igh t8 b e i n g p r o p o r t i o n a l t o t i le zezn i xcone of e a c h
lke e q u a t i o n (4 .5 .1 ) czn be u sed t o a n a l y z e - : , c o n t r i S u t i o n of i n s j t z ' i i t y
o f each f a c t o r income t o t h e t o t a l I n e q u a l i t y . / Tc i i l u s t r e t e thi t r n m r r i c a i 2 . a =e
. , ~ i l i ~ e t h e d a t a obLained from t h e A u s t r a l i a n Survey of C o n s u ~ e r Expenditure 2nd
Finance, 1967-68.L/ The r e s u l t 8 a r e p r e a e n t e d i n Table 2 . I t is seen f r o 5 :he
: ab le t h a t t h e income f rom employment, i . e . , wages and s s ia r ies c o n t r i b u t e 92.687
t o t h e t o t a l i n e q u a l i t y . Unincorpora ted b u s i n e s s i n c o ~ e ii; sccond con t r i b c t -np
L l . i8X nlid t h e p r o p e r t y income, i . e . i n t e r e s t , d iv idend n x d rant c o n t r i b u t e ~ 7 r r I v
3 . 2 4 1 t o t h e t o t a l i n e q u a l i t y .
- - - - -- - . . - - - -- *I ' i h i * srohlern h a s a l s o been c o n s i d e r e d b y , i a n i a - z 0 ! 2 j 2r.c ?-.- i i 5 1 . . - 7 ' See Prydder and Kakvani i 8 1 .
L
' T ? . t . dt . lv ir lJ r q u a t i o n s o f t h e l i n e a r c x p e n d i t u r c ~ y s t t - n (LES) a r e gi-;::
5 y
vi p l y i 7- b i ( v - a ) ( L . 6 . l )
w!~erc v i - piqi i s t h e p e r c a p i r a e x p e n d i t c r e f o r t h e L- th c o r z c d i t p , pi n
Is i t s p r i c e a n d q i i s t h e p e r c n p i t a q u a n t i t y derar.cle.!. v = 1 piqi i e i=l
n
L p i y i i s t h e s d b s i s t e n c e e x p e n d i t c r e . t o t a l p e r c a p i t a e x p e n d i t u r e and a - ' I= 1
E i 1s i n t e r p r e t e d as t h e m a r g i n a l b u d g e t s h a r e of t h d i t ! c o m o d i t y .
T h e a b o v e s y s t e m o f demand e q u a t i o n s i s d e r i v e d by r n x i m i z i n g t h e b
K l e i n a n d Rubln [ 4 ] form o f t h e u t i l i t y f u n c t i o n .
n u = B i l o g ( q t - yi ) ( L . 6 . 2 )
i-1 n
I n r h l c h t h e 13's and y ' s are p a r a m e t e l 8 w i t h 0 < E l 1, B j = 1 , yl 2 0 i= 1
and q i - y i > 0.
Let G i b e t h e G i n i - i n d e x f o r t h e d i s t r i b u t i o n of t h e e x p e n d i t u r e sc
t h e i - t t l conin~odi ty and G* be the (;!~il-index f o r t h e t o t d l e r p c n d i t u r e , their
u s l n p C o r o l l a r y 6 nn t h e e q u a t i o n ( 4 . 6 . 1 ) we o b t a i n
* w t ~ c r c u 1s t h e m a n t o t a l e x p e n d i t u r e and u i i~ t h c rneen e x p e n d i t u r e ;.: - ' i
t h c i - t h cnrrmodit?. T l i i s e q u a t i o n car1 a l s o be w r i t t e n ds
t:xper!r!it UILP c l . t s t j c i t y of t h e i - t h c c ~ n o d l t y d t t h e mc3a-1 t s i ~ c n d ! t u r e s t~ t-i;;uzl
t o c t ~ r r a i i o o f t h e Ginl-indices of tile d j s t r i b ~ l t i o n s ni tLlc I-t:? c o m c d i ~ : :
e x p e n d i t u r e s and t h e t o t a l e x p e n d i t u r e r e s p e c t i v e l y . If t h e e l a s t i c i . t y i s
g r e a t e r ( l e s s ) chan one, t h e e x p e n d i t u r e on t h e i - t h c o m o d i t y is more ( l e s s )
r ~ n e l u ~ l l v d i s t r i b u t e d t'!ian t h e t o t a l e x p e n d i t u r e .
4 . 5 . 1 Tnconc I n e q u a l i t y and Pr-
We now c o n s i d e r t h e e f f e c t of p r i c e changes on t h e income i n e q u a l i t y
of t h e r e a l income.
S u b s t i t u t i n g (4.6.1) i n t o ( 4 . 6 . 2 ) , we o b t a i n t h e i n d i r e c t u t i l i t y
f u n c t i o n a s
* Suppose t h e p r i c e s pi change t o pi , and t h e t o t a l e x p e n d i t u r e -J I
chenges t o v*, t h e n t h e r e e u l t i n g change i n t h e u t i l i t y w i l l be
n h *
where s - 1 p i y i . I f t h e change i n u t i l i t y i s a e t t o r e m , we o b t a i n t h e i-1 .
t o t a l p e r c a p i t a e x p e n d i t u r e v* i n o r d e r t h a t t h e f ami ly m a i n t a i n s t h e same
- - v4 w i l l b e t h e - r e e l e x p e n d i t u r e . Le t GR b e t h 8 ~ i n i - i n d e x of t h e r e a l e-rn- - - . d l t u r e , t h e n a p p l y C o r o l l a r y 8 on t h i e e q u a t i o n g i v e s
s -
. * --.'ere i s t h e G in i - indsx of t h e ncney e x p e n d i t u r e i n t h e base y e a r .
Tr i s obvious from t h e e q u a t i o n ( 4 . 6 . 8 ) t h a t i f a l l t h e p r l c e s c;.ange
i n :he same p r o p o r t i o n GR - G* i . e . , t h e i n e q u a l i t y of t h e d i s t r i b u t i o n of
t h e mozey e x p s n d i t u r e i n t h e b a s e y e a r is szme a s t h e i n 2 q u a l i t y of t h e r e a l
e x p e n d i t u r e .
* The r a t i o l- i s t h e t r u e c o s t of l i v i n g index.?j It c o n v e r t s t h e
v
money e x p e n d i t u r e i n t o real e x p e n d i t u r e . I n t h e s p i r i t o f t r u e c o ~ s t of l i v i r g
c.R index , we propoee t o u s e t h e r a t i o - u s an i ndex of t h e iccom~e i n e q u a l i t y G*
t o t a k e i n t o accoun t t h e e f f e c t s o f r e l a t i v e p r i c e changes . T i i ~ index c o n v e r t s
t h e i n e q u a l i t y of t h e money l iousehold e x p e n d i t u r e d i s t r i b u t i o n t o t h e i n e q u a l i t y
o f :he r e a l household e x p e n d i t u r e . I f t h i o i ndex i s l e s s t han one , i t i m p l i e s
t h a t t h e r e l a t i v e p r i c e changes a r e making t h e e x p e n a i t u r e d i s t r i b u t i o n more
i a e q u n l .
The numer i ca l r e s u l t s on t h e i n d e x o f ir.corne i n e q u a l i t y a r e p r e s e n t e d
i n Tab le 3. The U-K d a t a was used f o r t h i s purpose .? - / It i s eecn from t h e
t a b l e t h a t t h e r e l a t i v e p r i c e changee from 1964 t o 1972 have t h e e f f e c t o f
L i n c r e a s i n g income i n e q u a l i t y . The 1971-72 change f s p a r t i c u l a r l y a a r k c d .
4 . 6 . 2 Zrlcone I n e q u n l i t y and P r i c e s : An A l t e r n a t i v e Approach - - ' Z
Suppose t h e p r i c e of j - t t commodity changes by cl j p e r c e n t , t h e r f - - *
t h e dcmnnd f o r t h e i t commodity w i l l change by n i j a j percent:, v h e r e ) I -
r l i j i s t h e p r i c e e l a s t i c i t y of t h e i - t h c o m o d i t y w i t h r e s p e c t t o j - t h
p r l c e . -The r e s u l t i n g demand f o r t h e i - t h c o m o d i t y b e c o ~ e s
'/ See Kle ln and Rubin [ 4 1.
See X a e l l b a u e r [ 5 1 f o r t h e d e t a i l e d d e s c r i p t i o n of :b,c d t i r a .
Table 3: INDEX OF INCOME INEQUALITY IN U.K. 1964-72
.-, - - - T--
' --- - - - ,, r:hanpc I n Gin!- ' I i - , '~ '
- - --- Food 1 . 2 2 1
C l o t h i n g .037
I I !ousing --. 148 !
Durables - . l b 0 1 t
Others - 1 . 5 2 L I I I I
- ---- ----- -- ;
I C i h l e 6 g i v e s the p e r c e n t a g e c h a n g e 111 the Gini-j : :?~:: 5 : t>.t ril;,l P X ; ) ~ ~ Z -
d l t * rt. ~ I ' c l : thc p r i c e o f e a c h c o m o d i t y 11as Incrc;,stn4 t * . - ' .. .- ii t~~:e. 1s
!7cc.r. : l 4 . 3 t t h e p r i c e i n c r e a s e of foot- a n d c lo th i r rg i n c r c j t t . , r : , t incruaiity of
r{:,il e x r c n c i i t \ ~ r e w h i l e t he i n c r e a s e i n p r i c e of t h r e e oti,c.r g o c d s c e ; r a s e :kt
The e x p e n d i t u r e on t h e I - t h c o m d i t y a t base y e a r p r i c e s w i l l be
The t o t a l e x p e n d i t u r e is t h e n o b t a i n e d a s :
n where t h e u s e h a s been made o f t h e r e e t r i c t i o n 1 Si - 1 .
i- 1
Ad
L e t C;R b e t h e Gin i - index of t h e r e a l e x p e n d i t u r e , t h e n a p p l y h g
C o r o l l a r y on t h e e q u a t i o n (4.6.12) g i v e s
The e i rpreee ion (4.6.13) p r o . ~ i d e s t h e pe rcen tnge chnngc i n t h e Gin i - i r -dex
. of t h e r e a l e x p e n d i t u r e w h e n G h e p r i c e o f - t h e j - t h corvlaodity chsmgea by a + Z, 4
'2 o t h e r p r i c e s :remaining c o n s t a m . -
For t h e n u m e r i c a l i l l G t r a t l o n we used t h e d a t a o b t a i n e d from t h e !-lexica I
Household Survey conducted by t h e Bank o f Kexico in 1368. The f a a i l i e ~ con-
~ i d e r e d wcre urban e n t e r p r e n e u r s . The p a r a m e t e r s o f t h e l i n e a r e x p e n d i t u r e :;ye-
tern were s e t h t e d u s i n g i n d i v i d u a l o b s e r v a t i o n s ,
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